From Halley to Secant: Redefining root finding with memory‐based methods including convergence and stability

Abstract

Root‐finding methods solve equations and identify unknowns in physics, engineering, and computer science. Memory‐based root‐seeking algorithms may look back to expedite convergence and enhance computational efficiency. Real‐time systems, complicated simulations, and high‐performance computing demand frequent, large‐scale calculations. This article proposes two unique root‐finding methods that increase the convergence order of the classical Newton–Raphson (NR) approach without increasing evaluation time. Taylor's expansion uses the classical Halley method to create two memory‐based methods with an order of 2.4142 and an efficiency index of 1.5538. We designed a two‐step memory‐based method with the help of Secant and NR algorithms using a backward difference quotient. We demonstrate memory‐based approaches' robustness and stability using visual analysis via polynomiography. Local and semilocal convergence are thoroughly examined. Finally, proposed memory‐based approaches outperform several existing memory‐based methods when applied to models including a thermistor, path traversed by an electron, sheet‐pile wall, adiabatic flame temperature, and blood rheology nonlinear equation.